The goal of this paper is to establish the Lipschitz and W2,∞ estimates for a second-order parabolic PDE ∂tu(t,x)=12Δu(t,x)+f(t,x) on Rd with zero initial data and f satisfying a Ladyzhenskaya–Prodi–Serrin type condition. Following the theoretic result, we then give two applications. The first is to discuss the regularity of the stochastic heat equations, and the second is to discuss the Sobolev differentiability

We study an inverse problem for the first-passage place of a one-dimensional diffusion process X(t) (also with jumps), starting from a random position η∈[a,b]. Let be τa,b the first time at which X(t) exits the interval (a,b), and πa=P(X(τa,b)≤a) the probability of exit from the left of (a,b). Given a probability q∈(0,1), the problem consists in finding the density g of η (if it exists) such that πa=q

We propose a nonparametric estimation of random effects from the following small fractional diffusions dXi(t)=βib(Xi(t))dt+εdWi,H(t), Xi(0)=x0i, 0≤t≤T. For i=1,⋯,N, βi is random variable and (Wi,H(t), t≥0, i=1,⋯,N) is are standard fractional Brownian motions with a common known Hurst index H∈(1/2,1). The asymptotic behavior of the proposed estimators is established when ε and N are, respectively, sufficiently

We study the locally uniform convergence from a family of pullback random attractors to a deterministic attractor. We establish criteria by using joint convergence of the cocycles, collective local compactness and deterministic recurrence of the random attractors. All three conditions are verified in the content of stochastic non-autonomous Magneto-hydrodynamics equations under the weak assumption

The Adomian decomposition method (ADM) is a powerful tool to solve several nonlinear functional equations and a large class of initial/boundary value problems. In this paper, we discuss a probabilistic approach to compute the Adomian polynomials (AP’s), which is the main part of the ADM. We provide a probabilistic interpretation for the AP’s, both for the one-variable and the multivariable case. We

This paper deals with the limiting behavior of non-autonomous stochastic reaction-diffusion equations without uniqueness on unbounded narrow domains. We prove the existence and upper semicontinuity of random attractors for the equations on a family of unbounded (n + 1)-dimensional narrow domains, which collapses onto an n-dimensional domain. Since the solutions are non-uniqueness, which leads to a

We provide conditions implying finite-time blowup of positive weak solutions to the semilinear equation du(t,x)=[Δαu(t,x)+Ku(t,x)+u1+β(t,x)]dt+μu(t,x) dBtH,u(0,x)=f(x),x∈R d,t≥0, where α∈(0,2],K∈R,β>0,μ≥0 and H∈[12,1) are constants, Δα is the fractional power −(−Δ)α/2 of the Laplacian, (BtH) is a fractional Brownian motion with Hurst parameter H, and f≥0 is a bounded measurable function. To achieve

In this paper, we consider a stochastic tuberculosis model with two kinds of treatments, that is, treatment at home and treatment in hospital. Firstly, we obtain sufficient conditions for extinction and persistence in the mean of the diseases, then in the case of persistence, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions

Complementary regularity between the integrand and integrator is a well known condition for the integral ∫0Tf(r) dg(r) to exist in the Riemann-Stieltjes sense. This condition also applies to the multi-dimensional case, in particular the 2 D integral ∫[0,T]2f(s,t) dg(s,t). In the paper, we give a new condition for the existence of the integral under the assumption that the integrator g is a Volterra

In this article, we derive the state probabilities of different type of space- and time-fractional Poisson processes using z-transform. We work on tempered versions of time-fractional Poisson process and space-fractional Poisson processes. We also introduce Gegenbauer type fractional differential equations and their solutions using z-transform. Our results generalize and complement the results available

In the present paper we consider a family of non-Volterra quadratic stochastic operators depending on a parameter α and study their trajectory behaviors. We find all fixed and periodic points for a non-Volterra quadratic stochastic operator on a finite-dimensional simplex. A complete description of the set of limit points is given, and we show that such operators have the ergodic property.

The aim of this article is to consider the existence, uniqueness and uniformly exponential p-stability of the solution for ∇-delay stochastic dynamic equations on time scales via Lyapunov functions. This work can be considered as a unification and generalization of stochastic difference and stochastic differential time-varying delay equations.

In this work, we study the full randomized versions of Airy, Hermite and Laguerre differential equations, which depend on a random variable appearing as an equation coefficient as well as two random initial conditions. In previous contributions, the mean square stochastic solutions to the aforementioned random differential equations were constructed via the Fröbenius method, under the assumption of

In this article we define and investigate statistical operators and an entropy functional for Bernstein stochastic processes associated with hierarchies of forward-backward systems of decoupled deterministic linear parabolic partial differential equations. The systems under consideration are defined on open bounded domains D⊂Rd of Euclidean space where d∈N+ is arbitrary, and are subject to Neumann

In this paper, we consider volatility swap and variance swap when the underlying asset is described by a process with multiple stochastic volatility models. The model considered in this paper is the multi-factor Heston stochastic volatility model. We obtain pricing formulas for the weighted variance swap and approximate expressions for the weighted volatility swap. The bounds of the arbitrage-free

We are interested in the optimal filter in a continuous time setting. We want to show that the optimal filter is stable with respect to its initial condition. We reduce the problem to a discrete time setting and apply truncation techniques. Due to the continuous time setting, we need a new technique to solve the problem. In the end, we show that the forgetting rate is at least a power of the time t

In this paper, we study the dynamical behavior of a multigroup SIQS epidemic model, which is formulated as a piecewise deterministic Markov process. Sufficient conditions for extinction and persistence in the mean of the diseases are obtained. In addition, in the case of diseases persistence, we establish sufficient conditions for the existence of positive recurrence of the solutions to the model by

In this article, we study the following stochastic heat equation ∂tu(t,x)=Lu(t,x)+Ḃ, u(0,x)=0, 0≤t≤T, x∈Rd, where L is the generator of a Lévy process X in Rd, B is a fractional-colored Gaussian noise with Hurst index H∈(12, 1) in the time variable and spatial covariance function f which is the Fourier transform of a tempered measure μ. After establishing the existence of solution for the stochastic

In this paper, Wang’s Harnack inequalities and super Poincaré inequality for generalized Cox-Ingersoll-Ross model are obtained. Since the noise is degenerate, the intrinsic metric has been introduced to construct the coupling by change of measure. By using isoperimetric constant, some optimal estimate of the rate function in the super Poincaré inequality for the associated Dirichlet form is also obtained

Abstract The goal of this paper is to study a stochastic game connected to a system of forward-backward stochastic differential equations (FBSDEs) involving delay and noisy memory. We derive sufficient and necessary maximum principles for a set of controls for the players to be a Nash equilibrium in the game. Furthermore, we study a corresponding FBSDE involving Malliavin derivatives. This kind of

Abstract In this article, we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existing fully explicit space-time discrete approximation scheme, in particular

The main result of the paper deals with weak compactness in distribution of weak solutions sets to stochastic differential inclusions defined by set-valued stochastic integrals presented in the paper [Kisielewicz, M., Michta, M. (2017 Kisielewicz, M., Michta, M. (2017). Integrably bounded set-valued stochastic integrals. J. Math. Anal. Appl. 449(2):1892–1910. DOI: 10.1016/j.jmaa.2017.01.013.[Crossref]

Let ξ:Ω×Rn→R be zero mean, mean-square continuous, stationary, Gaussian random field with covariance function r(x)=E[ξ(0)ξ(x)] and let G:R→R such that G is square integrable with respect to the standard Gaussian measure and is of Hermite rank d. The Breuer-Major theorem in it’s continuous setting gives that, if r∈Ld(Rn), then the finite dimensional distributions of Zs(t)=1(2s)n/2∫[−st1/n,st1/n]n[G

This paper provides a theoretical analysis on the impacts of using a suboptimal information set for the estimation of the pricing kernel and, more in general, for the validity of the fundamental theorems of asset pricing. While inferring the risk-neutral measure from options data provides a naturally forward-looking estimate, extracting the real world measure from historical returns is only partially

In the present paper, we focus on a stochastic predator-prey model with stage structure for prey. Firstly, by using the stochastic Lyapunov function method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the predator population in two cases. Some

We study Brownian motion in the setting of quaternion analysis. We give a quaternion version of the Itô’s integral.

In this work, we develop polynomial processes where the coefficients of the process may depend on time. A full characterization of this model class is given by means of their semimartingale characteristics. While in the time-homogeneous case, moments can be calculated easily through matrix exponentials, we show that in the inhomogeneous case this is no longer possible. As an alternative we explore

The expected information gain is an important quality criterion of Bayesian experimental designs, which measures how much the information entropy about uncertain quantity of interest θ is reduced on average by collecting relevant data Y. However, estimating the expected information gain has been considered computationally challenging since it is defined as a nested expectation with an outer expectation

Oncolytic virotherapy is emerging as a promising new method for cancer treatment. In the present study, we propose and analyze a mathematical model for treatment of cancer by using oncolytic virotherapy. Here, it is assumed that the growth of tumor cells follows logistic growth and the interaction between tumor cells and viruses is of saturation type. Additionally, the removal rates of uninfected and

In this paper, we explore the stochastic viral infection model and show that this model has a unique global solution. We use the method of Lyapunov function to study the stochastic asymptotic stability of equilibrium solution for this model. Finally, the sufficient condition for persistence of the disease is established and our mathematical findings are illustrated.

We discuss the following problem: Given a set of information criteria for optimal designs, the numerical and computational complexity may drastically differ from one criterion to another. A general novel methodology, called the “FIRCEP” is introduced, and shown to work satisfactorily on a variety of problems relating weighted estimation criterion and Integrated mean square prediction error (IMSPE)

In this paper, we prove the existence and regularity of a tempered pullback attractor for a non-autonomous stochastic fractional equation involving an abstract non-local operator defined as LKu(x)=limε↘0∫RN∖Bε(x)(u(x)−u(y))K(x−y)dy,x∈RN, where K:RN∖{0}→(0,+∞) is the kernel of LK which satisfies the general fractional-type condition of order s,0

In this article, we study two stochastic predator-prey models with distributed delay and general functional response. For the nonautonomous periodic case of the system, by utilizing Khasminskii’s theory of periodic solution, we prove that the system admits a positive T-periodic solution. For the system which is perturbed by both white and colored noises, we establish sufficient conditions for the existence

In this paper the asymptotic behavior of multitype Markov branching processes with discrete or continuous time is investigated in the positive regular and nonsingular case when both the initial number of ancestors and the time tend to infinity. Some limiting distributions are obtained as well as multivariate asymptotic normality is proved. The paper considers also the relative frequencies of distinct

In this paper, the qualitative analysis of a stochastic two delayed epidemic model incorporating Lévy processes with a general non-linear incidence transmission is considered. Our analysis show that the proposed model is mathematically and biologically well-posed. Moreover, we develop a mathematical method to present the sufficient conditions for extinction and persistence in the mean of disease. Numerical